The normal boiling point of diethyl ether is \(34.5^{\circ} \mathrm{C}\). A solution containing a nonvolatile solute dissolved in diethyl ether has a vapor pressure of 698 torr at \(34.5^{\circ} \mathrm{C}\). What is the mole fraction of diethyl ether in this solution?

Vapor pressure is a term used to describe the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature in a closed system. This equilibrium occurs when the rate of evaporation of a liquid (or sublimation of a solid) equals the rate of condensation of its vapor.

Imagine a closed container partially filled with a liquid. Initially, some molecules at the surface of the liquid will have enough energy to escape into the gas phase, creating vapor. As the amount of vapor increases, some of these molecules will return to the liquid phase—this is condensation. When the number of molecules transitioning from liquid to vapor is equal to those condensing back to the liquid, equilibrium is reached.

In the context of the given problem, diethyl ether has a certain vapor pressure at its normal boiling point, 760 torr. When a nonvolatile solute is added, it disrupts the equilibrium and the vapor pressure over the solution changes. This change is crucial for calculating the mole fraction using Raoult's Law.

Mole fraction is a way of expressing the concentration of a component in a mixture. Mathematically, it is defined as the ratio of the number of moles of a particular substance to the total number of moles of all substances present in the mixture. The mole fraction, represented by the symbol \(X\), has no unit because it is a ratio of the same quantities. It is also always a number between 0 and 1.

For example, consider a mixture of diethyl ether and a nonvolatile solute. Here's how you would calculate the mole fraction of diethyl ether: \[X_{diethyl\ ether} = \frac{\text{number of moles of diethyl ether}}{\text{total number of moles in the solution}}\]

In our previous exercise, by re-arranging Raoult's Law, the mole fraction of diethyl ether could be found using the vapor pressure of the solution and the vapor pressure of pure diethyl ether. Essentially, the presence of the nonvolatile solute decreases the mole fraction of diethyl ether compared to its pure state.

The boiling point of a liquid is the temperature at which its vapor pressure equals the atmospheric pressure surrounding the liquid, resulting in the formation of bubbles within the liquid mass. In other words, it is the temperature at which a liquid turns into vapor.

A pure substance has a specific boiling point at a given atmospheric pressure. For diethyl ether, this temperature is \(34.5^\circ \mathrm{C}\) at 1 atmosphere of pressure. However, when a nonvolatile solute is dissolved in the liquid, this causes a phenomenon known as boiling point elevation. The solution's boiling point will be higher than that of the pure solvent because the solute particles interfere with the formation of bubbles by increasing the solution's overall vapor pressure. The amount of boiling point elevation is directly related to the number of solute particles present.

In the set exercise, understanding the boiling point of diethyl ether helps determine the vapor pressure at that temperature. It sets a reference point for calculating the mole fraction of diethyl ether in a solution using Raoult's Law. The problem made use of the boiling point to infer the vapor pressure of the pure diethyl ether, which was key to finding the solution's composition.